# how to prove ‘¬∃xP(x)→(P(a)→Q(a))’ from no premises? fitch

I am totally lost on how to do this... can anyone help?

What does it mean? I tried to understand what it means before proof but am totally clueless

• Assume the antecedent and assume Pa. From it, by exists-intro, derive a contradiction. – Mauro ALLEGRANZA Nov 22 '19 at 7:37

¬∃xP(x)→(P(a)→Q(a))

What does it mean? I tried to understand what it means before proof but am totally clueless

It says: `P(a)→Q(a)` is true, if `P(x)` holds for no `x`.

So why would `P(a)→Q(a)` be true when that is assumed?

@GrahamKemp is correct. The statement says that if P(x) holds for no member of the universe of discourse, then P(a)->Q(a).

The fact that we don't have a def for P(x) or Q(x) is is irrelevant because of the truth-table for the material conditional, which tells us that anytime the antecedent --here P(a)-- is false, the material conditional is true. [Remember, validity in natural logic is about truth preservation it can't help you when you start with a false statement].

So the proof is relatively easy: 